Compared to the well-known theory of stable populations, little was known about the dynamics of populations when fertility and mortality change arbitrarily over time until recently. While the result of weak ergodicity--that the age distribution "forgets" its past--was known, the dynamics of such populations were not specified mathematically. The proposed project is to provide a more complete formal analysis of the dynamics of populations with changing vital rates beyond the results obtained recently by Kim and Sykes. More specifically, the project aims to unify the two expressions for the relative age distribution recently obtained independently, and to examine the indirect estimation procedure using the new formulas when demographic data are defective without using the assumption of stability. Other aims are to discover how fast a population forgets its past history, how long it takes for the momentum of population growth to be specified and the determinants of these speeds of convergence. The recent decline of fertility and mortality in many countries and the resultant aging populations make the next aim to understand the dynamics of zero-growth populations and replacement-level fertility important. Because fertility and mortality are likely to adjust for the conditions in which a population is in, understanding the dynamics of such populations is the next aim. Extending the above research for closed populations to multistate populations is the next aim. "States" may be geographic regions (taking migration into account), marital statuses, or health statuses, to name a few. Understanding the dynamics of populations in as many states as necessary (in addition to the usual age dimension) will allow one to make better predictions of the future courses of populations. The research will be carried out in the discrete time formulation of one-sex population in n age groups and m regions (states) using population projection (or Leslie) matrices that change over time.